f f(xy)dx = ô(y)-x f fi(x)dx. - American Mathematical Society€¦ · MICHAEL COWLING, GARTH GAUDRY, SA VERIO GIULINI AND GIANCARLO MAUCERI Abstract. For a Lie group G with left-invariant

transactions of theamerican mathematical societyVolume 323, Number 2, February 1991

WEAK TYPE (1,1) ESTIMATES

FOR HEAT KERNEL MAXIMAL FUNCTIONSON LIE GROUPS

MICHAEL COWLING, GARTH GAUDRY, SA VERIO GIULINI ANDGIANCARLO MAUCERI

Abstract. For a Lie group G with left-invariant Haar measure and associated

Lebesgue spaces LP(G), we consider the heat kernels (pt}t>0 arising from a

right-invariant Laplacian A on G : that is, u(t, •) = pt* f solves the heat

equation (d/dt - A)u = 0 with initial condition u(0, •) = /(•). We establish

weak-type (1,1) estimates for the maximal operator J? (£f = sup(>0 \pt*f\)

and for related Hardy-Littlewood maximal operators in a variety of contexts,

namely for groups of polynomial growth and for a number of classes of Iwa-

sawa AN groups. We also study the "local" maximal operator J?q (J?0f =

suPo<«i 1^« * /I) anc^ related Hardy-Littlewood operators for all Lie groups.

1. Introduction

Let G be a connected Lie group, with fixed left-invariant Haar measure dx,

associated Lebesgue spaces LP(G), and modular function 5, so that

f f(xy)dx = ô(y)-x f fi(x)dx.Jg jg

As usual, we identify the Lie algebra g of G with the tangent space at the

origin, and denote by exp the exponential mapping. Let {Xx, ... , Xf} be a

basis for g and take k in [1, d] such that {Xx, ... , Xf} generates g as a Lie

algebra. Define the corresponding sub-Laplacean A to be the right-invariant

differential operator on C^°(G) given by the formula

k

àfix) = ^2(d/dt)2f(exp(tXj)x)\t=0 Vx g G ;;=i

A may also be considered as a densely-defined operator on LP(G).

We may associate to A a family of heat kernels {p,}t>0 such that

(i) ptGC°°(G) for all t in R+ ;

(ii) ô(x)pfx)=pt(x~x) and pt > 0 for all t in R+ ;

Received by the editors February 24, 1989.

1980 Mathematics Subject Classification (1985 Revision). Primary 22E30; Secondary 42B25.Research supported by the Australian Research Grants Scheme, the Italian Consiglio Nazionale

delle Ricerche, and the Italian Ministero della Pubblica Istruzione.

© 1991 American Mathematical Society

0002-9947/91 $1.00+ $.25 per page

6.17

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

638 MICHAEL COWLING, ET AL.

(iii) H/7,11, = 1 for all / in R+;

(iv) d/dtip, * fi) = A(p, * f) for all t in R+ and every / in LP(G), 1 <p <oc;

(v) limt^0+pt * fi = fi for every / in LP(G), where the limit is in the

Lp-norm if 1 < p < 4oo, and in the weak-star topology if p = co .

A relaxed account of the theory in the case where A is a Laplacean (i.e.

k = d) can be found in A. Hulanicki [7]. In the general case, some modifications

must be made to take into account the fact that A is subelliptic rather than

elliptic. However, using the theory developed by L. Hörmander [6], this is

straightforward.

In this paper, we consider the heat semigroup {.Pt},>0, where P = pt* . In

[11], E. M. Stein showed that the maximal operator JA associated to the heat

kernels

Jïfi = sup\pt*fi\ VfGLp(G);>0

is bounded on LP(G) provided 1 < p < oo. We show here that, if JAQ is the

"local" heat maximal operator

JAJ = sup \p,*fi\ VfcLp(G)0<f<l

then JA0 is of weak type ( 1, 1 ) for all connected Lie groups G, and that JA is

of weak type ( 1, 1 ) for many Lie groups, namely groups of polynomial growth

and certain Iwasawa AN groups. We also show that, in these cases, the heat

kernels resemble the heat kernels on R" ((4nt)~"' <?-'*' /4i) sufficiently closely

that we can prove weak (1,1) estimates for the Hardy-Littlewood maximal

operator obtained by averaging over "balls" in G.

The techniques we use to study maximal functions are simple, and we il-

lustrate them in the setting where G = R" and A is the standard Laplacean.

Since \\pt * fi\\p < \\fi\\p for all / in Lp(Rn) and for all p in [1, oo], theHopf-Dunford-Schwartz maximal ergodic theorem [2, p. 690] implies that, if

we define at (t in R+) and I? by the rules

at = \ft'psds

and

r/ = sup|fl,*/l VfGLp(Rn),t>0

then % is of weak type ( 1, 1 ). Now, for any x in R",

i \ 1 f2'i a \-n/2 -n/2 -\x\2/1s,at(x) = - (4n) s ' e ' ' ' ds

>w-^&jf!Af\-»'/-dst Jt

^ i * \-nß/~><\-"ß -kl2/4' i-«/2 / \> (4n) (2t) ' e ' ' ' =2 Pt(x).

It follows that JA is of weak type ( 1, 1 ).


HEAT KERNEL MAXIMAL FUNCTIONS 639

Finally,

pt(x) = (4n)-n/2rn/2e-M2/<1 > m-nl2t-nl2e-xX{u.]uH4t}(x)

> 2(ner(n/2))~x\{u G Rn: \u\2 < 4t}\~1 X{u:M^M)ix),

and so the weak type (1,1) boundedness of the Hardy-Littlewood maximal op-

erator, defined by taking averages over Euclidean balls, follows from the bound-

edness of the heat maximal operator J'.

In order to be able to estimate the heat kernels pt by (a multiple of) the

ergodic kernels at as above, we need a reasonably explicit formula for pt. The

general theory of the heat equation provides enough information to allow us to

treat the local maximal function JA^ on all Lie groups. On the other hand, the

estimates which are presently available (see e.g. S. Kusuoko and D. Stroock [9]

or N. Varopoulos [12]) allow us to establish the weak (1,1) boundedness of

the operator JA when G is of polynomial growth.

For the Iwasawa AN groups, we study a canonical Laplacean, viz one gener-

ated by an orthonormal basis of g and a basis of root vectors in n. We establish

a relationship between the heat kernels on AN and the heat kernels associated

with the Laplace-Beltrami operator ffA on the symmetric space G/K. (The

group G is assumed to have the Iwasawa decomposition G = A NK ; then

G/K = AN, as smooth manifolds.) Detailed information is available about the

heat kernels on G/K for large classes of symmetric spaces; e.g. N. Lohoué and

Th. Rychener [10] and J.-P. Anker and N. Lohoué [1]. Using their results and

the relationship between the heat kernels on AN (for A) and on G/K (for

A¿? ) we prove that JA is of weak type ( 1, 1) in the cases when G/K is of rank

1, G is complex, or g is a normal real form.

At an early stage of this work, we had established a special case of the re-

lationship between heat kernels on AN and those on G/K (Theorem 5.2).

We thank M. Flensted-Jensen for pointing out the result is general. The proofs

given in 5.1-5.2 are due essentially to him.

2. PRELIMINARIES

Suppose that G is a connected Lie group and Xx, ... , Xk generate g. De-

note also by Xx, ... , Xk the right-invariant vector fields agreeing with the re-

spective elements of the tangent space at e ; so

Xjfiix) = tLfiexpitXj)x)\t=0.

Define

d) A = Exr

The right-invariant operator A is selfadjoint and negative.



Associate a right-invariant distance d on G to {Xx, ... , Xk} as follows. Let

P be the set of all piecewise continuously differentiable mappings 7: [0, 1]-»G

such thatk

nt) = ^2akit)Xjiyit)) in Ty(l)(G),7=1

for all but finitely many t. Define the length of y , denoted \y\, by the rule

I*-/ (X>,«>l2) d,

Then define d by the formula

rf(x,y) = inf{|y|:yeP, 7(0) = x, y(l)=y} Vx,yeG.

Denote by B(x, r) the ball, centred at x, of radius r ; set |x| = d(x, e).

It can be shown that d is a metric on G which induces the usual topology,

and which comes from a Riemannian metric when Xx, ... , Xk span g. A

convenient reference for these facts is [12].

Consider now the heat diffusion boundary value problem in R+ x G :

<(d/dt-A)u = 0,

I «(*,■)-/(•) as?-0 + .

With the appropriate modifications to take care of the fact that {Xx, ... , Xk}

is not necessarily a basis of g, Theorem 3.4 of [7] shows that there is a unique

semigroup {Pt}t>0 of operators and a corresponding convolution semigroup

{pt}t>0 of kernels, with properties like those of the Gaussian kernels in the

Euclidean setting, such that, if f G LP(G) and u(t,-)=pt*f(-), then u isthe unique solution to (2), it being understood that the boundary value is taken

in the weak-star topology if p = 00 .

It is important for us to have precise information about the kernels of the

convolution semigroup {pt}[>0 , but this is difficult for arbitrary Lie groups G,

principally because different classes of Lie groups may behave in very different

ways at infinity. However, all Lie groups are locally Euclidean, so that useful

general local estimates can be proved. To state these, we denote by ß that

positive number for which r~fB(e, r)\ is bounded and bounded away from 0

for all r in (0,1]. See [12].

Theorem 2.1. Fix T in Rh. Then there are nonnegative constants C, C', and

C" such that

(a) whenever 0 < t < T and x G B(e, 1),

-Lt~ß/2exp(-\x\2/C't) <pt(x) < crß/2exp(-\x\2/C"t);

(b) whenever 0 < t < T and x G G\B(e, 1),

Ptix) < h(x)

where h(x) = 0(e~Mx{) for all X in R+ . In particular, h G LX(G).



Proof. These inequalities follow from the estimates of Kusuoka and Stroock [9]

or Varopoulos [12, §4]. D

3. Local maximal operators

Let the notation be as in §2.

Definition 3.1. For each measurable function / on G for which the following

formulae make sense, let

JA0f= sup \pt*fi\0<t<T

and

^fi= sup \B(e,t)\~x\x *fi\.0<t<T

The operators JAQ and ^ are called the local heat maximal operator and the

local Hardy-Littlewood maximal operator respectively.

Observe that the mapping properties of the operators Jt^ and ^ on the

Lp-spaces are independent of the choice of T. For ^, this is because, if

0 < a < b, then

sup \B(e,t)\~l\xB{e<t)*fi\ <\B(e,a)fXxB{e¡b)*\f\,

and the function \B(e, a)\~ Xßie ¿) is integrable. Similar reasoning applies to

JAQ.

Theorem 3.2. The operators JAQ and ^ are of weak type ( 1, 1 ).

Proof, (i) Consider first the operator JA0. By virtue of part (b) of Theorem

2.1,

(1) Pfx)<h(x)

whenever x £ G\B(e, I) and 0<t<T, where h G Lx (G).

On the other hand, if x G B(e, 1) and t < T/2, from part (a) of Theorem

2.1 we see that

(2) a,(x) = l- [ 'ps(x)dx > ±r(2t)-ßl2exp(-\x\2/C't) > Dpjx)1 Ji C

for appropriate positive constants D and X. From (1) and (2), there is a

positive constant E such that

pt(x) <E[h(x) + al-xt(x)] VxeG,

when t < T/2. Now by the Hopf-Dunford-Schwartz theorem, the maximal

operator f

IT:/i-» sup a,-.,* I/I0<t<T/2

is of weak type ( 1, 1 ). The operator /1-> h*f is of strong type ( 1, 1 ). Hence

JA0 is of weak type ( 1, 1 ).



(ii) To treat the operator ^ , we remark that, from part (a) of Theorem 2.1,

there are constants D and tí such that, if 0 < t < T and x G G,

rß/2XB(eS»)(x) ^ Drß/2exp(-\x\2/C't) < D'pt(x).

The weak (1,1) boundedness of ^ follows from that of JAQ and the earlier

remarks about the freedom in the choice of T. D

4. Groups of polynomial growth

Let G be a group of polynomial growth, i.e. such that for any relatively

compact neighbourhood U of e in G, | If" | grows polynomially in n. (We

recall that all Euclidean motion groups and all nilpotent groups have this prop-

erty.) Then there exists y in [0, +00), such that n~y\Un\ is bounded above and

away from 0 for all such neighbourhoods U [8]. In particular, in our general

framework, there are positive constants C and C1 such that

C<r~y\B(e,r)\<C' Vre[l,-K»),

and from [9] it follows that there are positive constants A, B, C,, C2, such

that

(1) Cxt~y/2exp(-\x\2/At) <pt(x) < C2t~y/2exp(-\x\2/Bt)

whenever t > 1 and x 6 G. See also [12].

Theorem 4.1. // G is of polynomial growth, the maximal operators JA and A%?

are of weak type (1, 1).

Proof. The boundedness of JA0 and ^ is taken care of in Theorem 3.2. The

proof of the boundedness of JA^ ,

'>1

follows the same lines as that of JAQ , once ( 1 ) is available.

The weak (1,1) boundedness of A?? is, as usual, a corollary of the result

for JA . D

5. Iwasawa .4 A groups

Let G denote a connected, noncompact semisimple Lie group and g its Lie

algebra. Denote by 6 a Cartan involution of g, and write

ß = eep

for the associated Cartan decomposition. Fix a maximal abelian subspace a of

p ; this determines a root space decomposition

g = g°e®ga,

aÇZ



X denoting the set of roots of the pair (g, a). Corresponding to a choice of

ordering of the roots, we have an Iwasawa decomposition

(1) g = í©a©n = É©a© 0 gQ;

further

g = mffia©nffiön,

m denoting the centraliser of a in É.

Write G = ANK for the Iwasawa decomposition of G corresponding to

(1). The solvable group AN is identifiable, as a manifold, with the symmetric

space G/K. Smooth functions on AN will be identified with smooth right

A"-invariant functions on G.

The Laplacean on AN. A distinguished Laplacean on AN can be constructed

as follows. Choose an orthonormal basis in a, say {Hx, ... , Hf and let

«.-ÍX;=iDenote by P+ the set of roots a of the complexification gc whose restriction

5 to a is nonzero. For each a in P. , choose X in g" such that X = 6X+ ' a v —a a

and

B(Xa,6Xß) = -20aß Va,/?eP+,

B denoting the Killing form on g.

Define the Laplacean A by setting

a = to + y x2.a / j a

a€P+

Proposition 5.1. (i) The Casimir operator on G is given by the formula

a = a -a -^(j i +x x ),a m 9 Z._à v a —a —a a' '

a€P+

where Qm is the sum of squares of the elements of a basis, orthonormal with

respect to -B, in m.

(ii) The Laplace-Beltrami operator S? for G/K acting on C°°(G/K) can be

identified with the operator Q. on C°°(G/K).

Proof. The first result follows from the definitions. Because Q is right-

invariant, Q maps C°°(G/K) to itself, and because Q comes from an Ad(G)-

invariant expression in the enveloping algebra of g, it is also left-invariant,

hence is identifiable with the Laplace-Beltrami operator. See also [4, p. 331]. D

The group AN has left Haar measure dadn where da and dn are the bi-

invariant measures on A and A respectively. The modular function 5 on AN

is given by the formula

S(an) = e-2p{Xosa] Man G AN,



p denoting one-half of the sum of the positive (restricted) roots, counted with

multiplicity. As remarked above, we shall also regard ô as a A^-right-invariant

function on G.

Theorem 5.2. (i) Suppose fGC°°(K\G). Then

SX/2AS-X/2f=Cif+(p,p)f,

where (■, ■) denotes the inner product dual to the Killing form on a*.

(ii) If fiG C°°(K\G/K), then

ôX,2AÔ-X/2fi = A?f+(p,p)fi

Proof, (i) Since / is invariant on the left under K,

(2) *V « O./ ~ J £ (*„*-. + X-«Xa)f-aeP+

Since X + X is in t, we have (X + X )/ = 0. Moreover, [X , X ] =a —a v a —a'J ' L a ' —aJ

-2H. , and so I X f - X X f = -2H.fi, since commutators of right-a ' —a aJ a —otJ aJ ' °

invariant vector fields correspond to minus the corresponding Lie bracket. Now

we see that

(XX +X X)f=2XX fi-2H.fi =-2X2 f-2H.fi.

It follows from (2) that

,2,Slf = Qaf+J£iHá + X¡)f

Now

(3)

a€P+

«a/+^/+E^2/=A/+//2/.a€r\

A(ô-x/2fi) = (na+J2xl)(s-l/2fi),

= Qa(ô-X,2fi) + S-X/2^xlfi,a€P^

since ô is A-invariant. For each j,

Hj(ô~X/2f)(ank) = p(H])ô~X'2(an)fi(ank) + â~X/2(an)Hjf(ank) ;

it follows that

H2(S~X/2fi) = p(Hj)2ô-X/2fi + 2p(Hj)Ô-X/2Hjf + ô-X/2H2fi,

and so

(4) na(â-x/2f) = (p, p)ô-x/2fi + â-x/2H2pfi + ô-x/2nj,

since the H commute. From (3) and (4) we conclude that

SX/2Aâ-X/2fi=(p,p)fi + H2pfi+ClJ+^2x2afi=Cifi+(p,p)fia€P+



(ii) If / is also right- K -invariant, then

by part (ii) of Proposition 5.1. D

Heat kernels on AN groups. We shall now exploit the connection between the

right-invariant Laplacean A on ^A and the Laplace-Beltrami operator Af? on

G/K, given in Theorem 5.2, to give an explicit expression for the heat kernels

associated to A.

Let {qt}t>0 be the heat kernels on G/K associated with f?. The func-

tions qt are regarded as rights-invariant functions on G, or by restriction as

functions on AN.

Theorem 5.3. The family {p,}l>0 given by the formula

,,, (p,p)t r—1/2(5) p, = e^ " ô qt

is the family of heat kernels for the operator A on AN.

Proof. The operator A is the infinitesimal generator of a C0 semigroup. As-

sociated with A there is a canonical semigroup of probability measures, say

{p't}t>0 [7]. We wish to show that pt = p\ for all t.

Fix / in C^(AN). The abstract Cauchy problem ACP2 with boundary

data / has a unique solution u [5, Theorem 23.8.1; see also p. 619], viz

u'(t,-)=p't*fi(-).

Once it is shown that, if

then u is also a solution to the same Cauchy problem, it will follow that, for

all t in R+.

P, * f = Pt * /

for all / in C^(AN), whence pt = p\.

It is shown in [7, Proposition 4.1] that, for every a in R+ and / in (0, R),

[ p't(x)eaMdx <C(a,R),JAN

where |x| denotes the Riemannian distance from e, canonically associated with

the choice of basis of g. Let 5 be negative, and consider the weight function

Ws

Ws(x) = esM vxeG.



Denote by P't the convolution operator defined by p\ (t > 0). Then if / G

CfAN) and 0 < t < R,

\\P'tf\\L\w)= f \P,*fix)\wsix)dx' J AN

< [ [ \p't(y)fi(x)\ws(yx)dxdyJan Jan

<ff p't(y)e-sMes[xl\fi(x)\dxdyJan Jan

<C(-s,R)\\fi\\L¡{Ws).

Since P't clearly carries L°°(AN) into L°°(AN), it follows by interpolation

that {P't},>0 can be thought of as a semigroup of operators on L (wf . The

infinitesimal generator of {P',}t>0 is A. Moreover, {P1} isa CQ-semigroup.

Consider a fixed / in C^°(AN), and the abstract Cauchy problem ACP2 with

values in L2(wf ; see Hille-Phillips [5, pp. 619-622]. We shall show in (i)-(iii)

that the function u(t, •) =pt* /(•) is a solution to this problem, provided 5 is

sufficiently large negative, whence by Theorem 23.8.1 of [5], it will follow that

Pt* fi = p't* f for all t in R+ .(i) u satisfies the heat equation on (0, oo) x AN.

(ii) «(*,.)-»/(•) in L2(wf as í-> 0+.(iii) u is absolutely continuous with continuous derivative on each compact

subinterval of R+ provided 5 is sufficiently large negative. To see this, write,

using the inversion formula for the spherical transform,

Pt = e{p'p)tô-XI2qt = C f e-t(X'l)[ö-xl2(p,]\c(X)\-2dX,Ja'

where C denotes the normalisation constant [4, Theorem 7.5]. So, by the mean

value theorem, if h > 0,

\\pt+h-pt\\ < Ch [{I + (X, X)}e-'{X'X)\\ô-x/2tpf\ \c(X)f2dX.Ja"

However, the spherical functions tpx are uniformly bounded for X in a , and

S~xt G L (w ) if 5 is sufficiently large negative. It then follows that

(6) \\Pt+h-Pt\\L2{w¡)<C(t)h,

where C(t) is continuous in /.

Now pass to u(t, •) = pt * /(•). We have

\u(t + h, ■) - u(t, -)| = f{pl+hi-y)-P,i-y)}fiy~l)dyJs

S denoting the compact support of /. For each y in S,

(7) \\p,+hi-y) - pti-y)\y{ws) $ c\\pt+h -pt\\LHw,)>

whereC = sup \S(z~x)w_(z)\x/ .

zes



It follows directly from (6) and (7) that the function / >-► u(t, •) satisfies the

stated absolute continuity condition. The derivative condition is established

along similar lines. D

From now on we shall assume that the symmetric space G/K satisfies one

of the following conditions:

1. G/K has rank one.

2. G is a complex group.

3. G is a normal real form.

Proposition 5.4. Let {pt}t>0 be the convolution semigroup of heat kernels (5).

There exist a measure space (B ,3S, p), nonnegative measurable functions tp,

\px, ... ,\pk on AN x B, and positive constants ax, ... ,ak such that

;=i

where

p?ix) = n I e-^y)"Wj(x,y)dp(y)J B

for all x in AN and t in R+.

Proof. 1. The case where G/K has rank 1. Then G/K can be realised as the

«-dimensional hyperbolic space //„(F) over F, where F = R, C, H or O,

i.e. the set of real numbers, of complex numbers, of quaternions or of Cayley

octonions. We write d = dimRF and I = nd ; I is the dimension of //„(F)

over R. Let a and 2q denote the positive roots with multiplities mx and m2

respectively. Choose Ha in a such that ot(Hf) = 1. Write

D = —L-Ix sinh x dx

The inversion formulae for the Abel transform on //„(F) [10, Hilfssatz 1] yield

the following expressions for qfexpxHf , for any x in R+ :

(a) If F = R and / is odd (i.e. d = 1, l = n is odd)

(8) <7,(exp(x//Q)) = ct-U2e-{p,P)tD(i-me-^iM_

(b) If F = R and / is even,

(9)f+-00 j

ql(expxHa) = crx,2e~{p'p)' / (coshy - coshx)"'72^2^ /4')sinhydy.J X

(C

(10)

c) If F = C, H or O, there exist positive constants cx, ... , cdj2 such that

dß ,+ooJ+l-d

q,(exp(xHa)) = ct x/2e <",">,¿c-/" °°(cosh2^ - cosh2x)"1/2(coshj)y,=i Jx

, r.j+d(n-l)/2 -v2/<^ . , ,x (EP " e ' )smhydy.



We define Or, for r in N, inductively by the formulae ®x(x) = x/ sinhx and

*r+lix) = Dx*r(x)

for all x in R. Then a simple induction argument shows that, for every positive

integer m ,, m ,

„m -x /It v^ i \*~J -x IMDxe = ¿^Wjix)t e

;=i

where the functions ipj are linear combinations, with positive coefficients, of

products <P, ■••<&. . Since

®r(x) = tpf+X)(exp(xHa)),

where tp^ r+ is the spherical function on H2r+X(R) corresponding to the eigen-

value -(p, p), the functions y/¡ are positive. The desired expression for pt

then follows from the respective formulae (8)—(10) in the manner indicated

below.

If d = 1 and n is odd (case (a)), simply take the one-point measure space

for (B, dp).If d = 1 and n is even (case (b)), take (B, dp) = (R+, dy) and perform

the change of variables u = cosh}' - coshx.

Similarly, if d > 1 (case (c)), take (B, dp) = (R+, dy), and make the1 1

change of variables u = cosh y - cosh x.

2. The complex case. The heat kernel for G/K is given by

(14) ql(x) = Crn'2e-{p>p)'ip0(x)e-M2/4'

for x in G/K [3]. Since the spherical function ç>0 is positive, the desired ex-

pression for pt follows from (14) if we take for (B, dp) the one-point measure

space.

3. Normal real forms. If the Lie algebra g of G is a normal real form

of a complex Lie algebra g, the method of reduction to the complex case of

Flensted-Jensen [3] (see also [1]) yields

;i5) qfexpH) = C / qf/2(kexp(H/2)) dkJ A

for all H in a, where qt is the heat kernel for the complex case, and K

is an analytic subgroup of G, the analytic group whose Lie algebra is g. Let

p and p be half the sum of the positive roots of the pairs (g, a) and (g, a)

respectively, and let ( , ), ||-||, ( , )_ , and ||-|L be the inner product and

norm induced by the Killing forms on the respective Cartan subalgebras a and

5. Since g is a normal real form of g, one has a = à. Moreover p = 2p and

2||//||i = ||//||2 for all H in a. Thus, by (14) and (15),

qt(expH) = C(t/2)-n/2e-{p'p)t í p0(£exp(///2))e-|¿exp("/2)|2/4'dk.Jkc



Once again, (5) yields the desired expression for pt. o

We come now to the maximal function theorem.

Theorem 5.5. For each of the three classes of groups G described before the state-

ment of Proposition 5.4, the maximal operator JA for the heat kernels {p,}t>0

associated to the Laplacean A on AN is of weak type (1, 1).

Proof. Recall the notation that at = f~ f ' Psds. By Proposition 5.4,t

<*iJJ)at(x)>a{t]) = rX j pfds>Ta>p

for j = I, ... , k. It follows from the Hopf-Dunford-Schwartz theorem that

each of the maximal operators JA¡,

JAf = sup\p{p*f\,3 «>o

is of weak type ( 1, 1 ). So therefore is J?. D

References

1. J.-Ph. Anker and N. Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J.

Math. 108(1986), 1303-1354.

2. N. Dunford and J. T. Schwartz, Linear operators, Vol. I, Wiley, New York, 1957.

3. M. Flensted-Jensen, Spherical functions on a real semisimple Lie group. A method of reduc-

tion to the complex case. J. Funct. Anal. 30 (1978), 106-146.

4. S. Helgason, Groups and geometric analysis, Academic Press, London, 1984.

5. E. Hille and R. S. Phillips, Functional analysis on semi-groups, Amer. Math. Soc. Colloq.Publ., vol. 31, Amer. Math. Soc, Providence, R.I., 1957.

6. L. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147-171.

7. A. Hulanicki, Subalgebra of LX(G) associated with Laplacian on a Lie group, Colloq. Math.31 (1974), 259-287.

8. J. W. Jenkins, A characterization of growth in locally compact groups, J. Funct. Anal. 12

(1973), 113-127.

9. S. Kusuoko and D. Stroock, Long time estimates for the heat kernel associated with a

uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), 165-189.

10. N. Lohoué and Th. Rychener, Die Resolvente von A auf symmetrischen Räumen vonnichtkompakten Typ, Comment. Math. Helv. 57 (1982), 445-468.

11. E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley Theory, Ann. ofMath. Stud., no. 63, Princeton Univ. Press, Princeton, N.J., 1970.

12. N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410.

School of Mathematics, University of New South Wales, Kensington N.S.W. 2033,Australia (Current address of Michael Cowling)

School of Mathematical Sciences, The Flinders University of South Australia,

Bedford Park S.A. 5042, Australia (Current address of Garth Gaudry)

Dipartimento di Matemática, Università di Genova, 16132 Genova, Italy (Current ad-

dress of Saverio Giulini and Giancarlo Mauceri)


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f f(xy)dx = ô(y)-x f fi(x)dx. - American Mathematical Society€¦ · MICHAEL COWLING, GARTH GAUDRY, SA VERIO GIULINI AND GIANCARLO MAUCERI Abstract. For a Lie group G with left-invariant